Is it possible to make a non-animated rope for game engine?
Separation


split the rope into hanging part (maybe two or more)
winch part
hook part (if even needed)


Now you can care each part individually. As you already wrote: it just needs to look like.

Hanging Rope

Assuming the rope is always straight:


small cylinder with origin at one of the caps
scale along it's length axis to change length
trackTo actuator to let it connect to a target object


Assuming the rope bends

Assumption: binding with less detail (no loops and high detailed waves etc.).


armature with a certain number of bones (e.g. 10)
skin mesh as cylinder (with enough vertices to deform with the bones)


Now you can use fixed actions or you manipulate the armature (e.g. via IK) get the animation you want. Be aware this does not consider any physics. Your animation technique should make it look like physics is involved.

You can use some wort of ragdoll methods. (You have an invisible physics model and copy the result over to the visible rope). As you said this is not the best option due to stretching.